Note that for triangle-free graphs \(dim_R(G) = \Delta (G)\) . Moreover, it can be shown (see [1]}) that if triangle-free graph \(G\)  is not a disjoint union of edges \(nK_2\) , then \(dim_P(\overline{G}) = \chi ^{\prime }(G)\) , where \(\chi ^{\prime }(G)\)  is a chromatic index. Therefore triangle-free fractals, that are not disjoint unions of edges, are exactly triangle-free graphs of class 2. On the other hand, all bipartite graphs except \(nK_2\)  are not fractals, since they all are traingle-free graphs of class 1.
